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Windowing Regularization Techniques for Unsteady Aerodynamic Shape Optimization

Steffen Schotthöfer, Beckett Y. Zhou, Tim Albring, Nicolas R. Gauger

TL;DR

This work tackles the challenge of sensitivity analysis for time-averaged objectives in URANS flows with limit-cycle oscillations and unknown period lengths. It introduces windowed time averages $J_w$ and their sensitivities, embedded within a Lagrangian-discrete adjoint framework in the SU2 solver, to regularize and stabilize optimization. The study shows that high-order windows (e.g., Hann, Hann-Square, Bump) achieve faster and more robust convergence of both objectives and sensitivities compared to the Square window, validating consistency between forward and adjoint modes and yielding stationary, lower-drag designs in NACA0012 and pitching-airfoil cases. The approach offers a practical, cost-effective path to robust unsteady aerodynamic shape optimization under periodic and potentially chaotic dynamics, with clear guidance on window choice for improved reliability and performance.

Abstract

Unsteady Aerodynamic Shape Optimization presents new challenges in terms of sensitivity analysis of time-dependent objective functions. In this work, we consider periodic unsteady flows governed by the URANS equations. Hence, the resulting output functions acting as objective or constraint functions of the optimization are themselves periodic with unknown period length, that may depend on the design parameter of said optimization. Sensitivity Analysis on the time-average of a function with these properties turns out to be difficult. Therefore, we explore methods to regularize the time average of such a function with the so called windowing-approach. Furthermore, we embed these regularizers into the discrete adjoint solver for the URANS equations of the multi-physics and optimization software SU2. Finally, we exhibit a comparison study between the classical non regularized optimization procedure and the ones enhanced with regularizers of different smoothness and show that the latter result in a more robust optimization.

Windowing Regularization Techniques for Unsteady Aerodynamic Shape Optimization

TL;DR

This work tackles the challenge of sensitivity analysis for time-averaged objectives in URANS flows with limit-cycle oscillations and unknown period lengths. It introduces windowed time averages and their sensitivities, embedded within a Lagrangian-discrete adjoint framework in the SU2 solver, to regularize and stabilize optimization. The study shows that high-order windows (e.g., Hann, Hann-Square, Bump) achieve faster and more robust convergence of both objectives and sensitivities compared to the Square window, validating consistency between forward and adjoint modes and yielding stationary, lower-drag designs in NACA0012 and pitching-airfoil cases. The approach offers a practical, cost-effective path to robust unsteady aerodynamic shape optimization under periodic and potentially chaotic dynamics, with clear guidance on window choice for improved reliability and performance.

Abstract

Unsteady Aerodynamic Shape Optimization presents new challenges in terms of sensitivity analysis of time-dependent objective functions. In this work, we consider periodic unsteady flows governed by the URANS equations. Hence, the resulting output functions acting as objective or constraint functions of the optimization are themselves periodic with unknown period length, that may depend on the design parameter of said optimization. Sensitivity Analysis on the time-average of a function with these properties turns out to be difficult. Therefore, we explore methods to regularize the time average of such a function with the so called windowing-approach. Furthermore, we embed these regularizers into the discrete adjoint solver for the URANS equations of the multi-physics and optimization software SU2. Finally, we exhibit a comparison study between the classical non regularized optimization procedure and the ones enhanced with regularizers of different smoothness and show that the latter result in a more robust optimization.

Paper Structure

This paper contains 11 sections, 1 theorem, 32 equations, 16 figures, 1 table.

Table of Contents

  1. Introduction
  2. Computing Sensitivities of Limit Cycle Oscillations
  3. The Optimization Framework
  4. Periodic detached flow around the NACA0012 airfoil
  5. Validation of the windowing approach
  6. Validation of the adjoint solver
  7. Comparison of surface sensitivities using different windows
  8. Comparison of optimization results using different windows
  9. Periodically pitching Airfoil
  10. Comparison of windowed time averages and windowed time averaged sensitivities
  11. Conclusion

Key Result

Theorem 1

The windowed time average ${J}_w$ and the windowed time averaged sensitivity $\frac{\;\mathrm{d}}{\;\mathrm{d} \sigma}{J}_w$ computed with a window $w( s)\in C^l(\mathbb{R},\mathbb{R})$ spanning $k = \lceil\frac{M}{T}\rceil$ periods converges to ${J}$ with order where For $f\in C^\infty(\mathbb{R},\mathbb{R})$, we get an exponential rate of convergence.

Figures (16)

  • Figure 1: Velocities of a flow around NACA0012 airfoil with $Re = 10^6$, $17$ degrees AoA at different time points of a period. The period length is $T(\sigma)\approx 31 \Delta t$.
  • Figure 2: Long-time behavior of $\frac{\;\mathrm{d}}{\;\mathrm{d} \sigma} C_D(n)$ over $n$; higher Reynolds numbers, i.e. more turbulent flows lead to exponential growths in the period amplitude of the sensitivity
  • Figure 3: Square- , Hann-, Hann-Square-, and Bump-window over $s$
  • Figure 4: $C_D(n)$ and $\frac{\;\mathrm{d}}{\;\mathrm{d} \sigma} C_D(n)$ over $n$ with $Re=10^6$; the window for the adjoint run spans iteration $500$ to $1200$
  • Figure 5: $C_D(n)$ long-time behavior with different Reynolds numbers. Note the slight shift upwards of the period mean in both cases, that stops at approximately $n=4500$.
  • ...and 11 more figures

Theorems & Definitions (1)

  • Theorem 1